On the Existence, Uniqueness, and Basis Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a Ball

We consider the following eigenvalue problem: − Δ ð ‘¢ + ð ‘“ ( ð ‘¢ ) = 𠜆 ð ‘¢ , ð ‘¢ = ð ‘¢ ( ð ‘¥ ) , ð ‘¥ ∈ ð µ = { ð ‘¥ ∈ â„ 3 ∶ | ð ‘¥ | < 1 } , ð ‘¢ ( 0 ) = ð ‘ > 0 , ð ‘¢ | | ð ‘¥ | = 1 = 0 , where ð ‘ is an arbitrary fixed parameter and ð ‘“ is an odd smooth function. First, we prove that for each integer ð ‘› ≥ 0 there exists a radially symmetric eigenfunction ð ‘¢ ð ‘› which possesses precisely ð ‘› zeros being regarded as a function of ð ‘Ÿ = | ð ‘¥ | ∈ [ 0 , 1 ) . For ð ‘ > 0 sufficiently small, such an eigenfunction is unique for each ð ‘› . Then, we prove that if ð ‘ > 0 is sufficiently small, then an arbitrary sequence of radial eigenfunctions { ð ‘¢ ð ‘› } ð ‘› = 0 , 1 , 2 , … , where for each ð ‘› the ð ‘› th eigenfunction ð ‘¢ ð ‘› possesses precisely ð ‘› zeros in [ 0 , 1 ) , is a basis in ð ¿ ð ‘Ÿ 2 ( ð µ ) ( ð ¿ ð ‘Ÿ 2 ( ð µ ) is the subspace of ð ¿ 2 ( ð µ ) that consists of radial functions from ð ¿ 2 ( ð µ ) . In addition, in the latter case, the sequence { ð ‘¢ ð ‘› / ‖ ð ‘¢ ð ‘› ‖ ð ¿ 2 ( ð µ ) } ð ‘› = 0 , 1 , 2 , … is a Bari basis in the same space.